# Definition:Totally Bounded Metric Space/Definition 1

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## Contents

## Definition

A metric space $M = \left({A, d}\right)$ is **totally bounded** if and only if:

- for every $\epsilon \in \R_{>0}$ there exists a finite $\epsilon$-net for $M$.

That is, $M$ is **totally bounded** if and only if:

- for every $\epsilon \in \R_{>0}$ there exists a finite set of points $x_1, \ldots, x_n \in A$ such that:
- $\displaystyle A = \bigcup_{i \mathop = 1}^n B_\epsilon \left({x_i}\right)$

- where $B_\epsilon \left({x_i}\right)$ denotes the open $\epsilon$-ball of $x_i$.

That is: $M$ is **totally bounded** if and only if, given any $\epsilon \in \R_{>0}$, one can find a finite number of open $\epsilon$-balls which cover $A$.

## Also known as

A **totally bounded metric space** is also referred to as a **precompact space**.

## Also see

- Results about
**totally bounded metric spaces**can be found here.

## Sources

- 1970: Lynn Arthur Steen and J. Arthur Seebach, Jr.:
*Counterexamples in Topology*... (previous) ... (next): $\text{I}: \ \S 5$: Complete Metric Spaces - 1975: W.A. Sutherland:
*Introduction to Metric and Topological Spaces*... (previous) ... (next): $7.2$: Sequential compactness: Definition $7.2.10$